[math-fun] mis-defining limits

I'm teaching an honors calculus class this semester, and to motivate the formal epsilon-delta definition of limits, I'd like to show the students some appealing, informal definitions of limits and the hidden errors and ambiguities of those definitions. Do any of you have any favorite examples of this? E.g.,, I seem to recall that some definitions of limits say that f(x) should approach but never equal b as x approaches a (which would make lim_{x -> 0} x sin x problematical). Jim Propp